Shehata E. Abdel Raheem , Yasser Abdel Shafy Fayez K ...Shehata E. Abdel Raheem, Yasser Abdel Shafy, 68 Fayez K. Abdel Seed and Hamdy H. Ahmed gravity load and additional lateral loads

67

Journal of Engineering Sciences, Assiut University, Vol. 41 No 1 pp. 67-88- January 2013

PARAMETRIC STUDY ON NONLINEAR STATIC ANALYSIS

OF CABLE STAYED BRIDGES

Shehata E. Abdel Raheem 1

, Yasser Abdel Shafy 2

, Fayez K. Abdel

Seed 3

and Hamdy H. Ahmed 4

1- Assoc. Prof. of Earthquake, Structural and Geotechnical Engineering, Taibah

University, KSA. On leave; Faculty of Engineering, Assiut University.

[email protected]

2 -Structural Engineer, Petroleum Projects and Technical Consultations Company

PETROJET

3- Prof. of Structural Engineering, Faculty of Engineering, Assiut Uni versity.

4- Assoc. Prof. of Structural Engineering, Faculty of Engineering, Assiut

University.

(Received August 25, 2012 Accepted 31 October , 2012)

Abstract

This study is done to discuss nonlinear static behavior for cable-stayed

bridges, hence develop a set of consistent design as well as a feasibility study

of long span cable-stayed bridges over Nile River. In order to accomplish

this goal, a thorough investigation of important key design parameters to

determine the behavior of cable-stayed bridge and identify any gaps in

current knowledge is done to be filled in order to enable the formation of a

consistent set of design recommendations. Three span cable stayed bridge

has been analyzed, the effects of the variation of different key design

parameters: cross section of cables, cable layout either fan or harp pattern,

pylon height to span ratio and mechanical properties of deck and pylon on

the straining action of the bridge elements are investigated. The loads on the

cable stayed bridge are a symmetrical load such as the own weight of all

structural elements and live loads. The results related to the major factors to

choose the ratio between the bending stiffness of a deck and axial stiffness of

the cable to reduce bending moments and deflections in the deck and pylon

are presented and discussed. Finally, some conclusions related to the cable

stayed bridge’s analysis/design are drawn.

Keywords: Cable-stayed bridge, Pylon, Nonlinear static analysis, Finite

element analysis, Design guidelines.

1. INTRODUCTION

As demands for improved infrastructure increase around the world, civil engineers

continue to be challenged to develop large bridges that must perform well even under

extreme loading. An effective means of bridging large distances in both seismic and

non-seismic regions is through the use of cable-stayed bridges. The decks of a cable-

stayed bridge are supported using cables that climb diagonally to strong stiff towers,

which act as the main load-bearing elements for the bridge. The orientation and

construction methodology adopted for the bridges is such that under uniform loading

the static horizontal forces imposed by the cables on the decks are typically balanced.

Consequently, the towers will be designed to resist the vertical component of the

Shehata E. Abdel Raheem, Yasser Abdel Shafy,

68 Fayez K. Abdel Seed and Hamdy H. Ahmed

gravity load and additional lateral loads associated with live loads, wind and seismic

actions, impact from colliding objects, drag from water flow, and possibly others. The

structural form and detailed behavior of cable-stayed bridges are well understood and

there is a considerable amount of literature on cable-stayed bridges [1 4].

Cable-stayed bridges can be a very effective means of bridging large distances in

both seismic and non-seismic regions. The rapid progress of this kind of bridges is

mainly due to the development of computer technology, high strength steel cables,

orthotropic steel decks and construction technology. Because of its aesthetic appeal,

economic grounds and the ease of erection, the cable-stayed bridge is considered as

most suitable for medium to long span bridges with spans ranging from 200 to about

1000 m [2, 5, 6]. As a matter of fact, the longer span length is, the more flexible the

bridge structural system behaves. Because of their huge size and complicated nonlinear

structural behaviors, the analysis of cable-stayed bridges is much more complicated

than that of conventional bridges. Their design, analysis, and construction can be very

challenging, and fortunately there is a considerable amount of literature that can assist

engineers with both the analysis and detailed design of cable-stayed bridges. It is less

common, however, to find simple recommendations for the conceptual design of cable-

stayed bridges, the sources of nonlinearity in cable-stayed bridges mainly include the

cable sag, beam-column and large deflection effects.

Gravity loading on the bridge will have a strong influence on the peak compression

forces that develop in the longitudinal axis of the deck. Therefore, one might consider a

strong heavy deck to resist the compression but of course the gravity loads themselves

are a function of the weight of the deck. In addition, while a heavy deck may not be

problematic for wind loading, it would certainly be an issue for seismic loading. As

such, the ideal deck section is a strong but lightweight deck. One might also consider

the benefits of a flexural stiff deck for non-seismic loads. An extensive parametric

study undertaken by Walther [3] showed that for static loads, the use of a stiff deck

section is not ideal for cable-stayed bridges, since it attracts significant bending

moments at three critical zones: deck-to-pier, abutments, and mid spans [7]. However,

this observation comes from static considerations, and a stiff deck can instead be quite

beneficial when dynamic wind and earthquake loads are considered. Also note that

vertical displacements of the deck may be most significantly affected by the stiffness

of the cable-pier system. This can be appreciated simply considering the typical span-

to-depth ratios of decks in cable-stayed bridges, which tend to be in the range of 100–200, well above normal ratios used to control deformations associated with beam

flexure. As such, the deck stiffness will be more relevant for local deformations

between cable support points and for dynamic vibrations associated with wind and

seismic response.

The spacing of cables should be set with due regard to construction lifting and

transport requirements for the deck, in addition to limiting static flexural demands. The

gravity loads are also likely to impose the greatest axial loads on the piers and

foundations. However, as both the piers and foundations of cable-stayed bridges tend

to be massive structures in order to provide adequate lateral stiffness for wind,

PARAMETRIC STUDY ON NONLINEAR STATIC ANALYSIS.... 69

earthquake, and eccentric gravity loading, the vertical loading of these elements is not

usually critical.

A long-span cable-stayed bridge exhibits nonlinear characteristics under loadings. It

is well known that these long-span cable-supported structures are composed of

complex structural components with high geometric nonlinearities: The nonlinear axial

force-elongation behavior for the inclined cable stays under different tension load

levels due to the sag initiated by their own weight (sag effect); The combined axial

load and bending moment interaction for the girder and towers; Large displacement,

which is produced by the geometry changes of the structure. In addition, nonlinear

stress-strain behavior of each element including yielding should be included in the

nonlinear analysis and overall safety evaluation.

The objective of this research is to study the static behavior of a cable stayed bridge

as well as a feasibility study of a long span cable stayed bridge. A reference model is

designed and used to investigate the influence of key design parameters on static

behavior of a cable stayed bridge. This model is submitted as part of a feasibility study

for a cable stayed bridge to cross over the Nile River, Egypt. The reference design is

modeled in a finite element program to investigate and calculate the force distribution

and deformations. With the help of the reference design, Three span cable stayed

bridge has been analyzed, the effects of the variation of different key design

parameters: cross section of cables, cable layout either fan or harp pattern, pylon

height to span ratio and mechanical properties of deck and pylon on the straining action

of the bridge elements are investigated. The loads on the cable stayed bridge are a

symmetrical load such as the own weight of all structural elements and live loads.

Finally, some conclusions related to the analysis/design of the cable stayed bridges are

drawn. The results of the parameter study are used to determine an optimization of the

reference design.

2. NONLINEAR CONSIDERATIONS IN ANLAYSIS

A cable-stayed bridge is a nonlinear structural system in which the girder is

supported elastically at points along its length by inclined cable stays. Although the

behavior of the material is linearly elastic, the overall load-displacement response may

be nonlinear under normal design loads [6, 8, 9]. Geometric nonlinearities arise from

the geometry changes that take place as the bridge deforms under loadings. As

mentioned above, there are usually three sources of the geometric nonlinearity: cable

sag effects; axial force and bending interactions; and large displacements. To account

for the sagging of inclined cables, an equivalent straight chord member with an

equivalent modulus of elasticity is considered [10].

22 12/)(1 TwLAE

EEeq

(1)

Where Eeq = equivalent elastic modulus of inclined cables; E = cable material

effective elastic modulus; L = horizontal projected length of the cable; w = weight per

unit volume of the cable; and T = cable tensile stress. The nonlinear analysis of a long-

span cable stayed bridge finally reduces to forming the nonlinear incremental

equilibrium equations of the system and to solving these equations. Based on the



characteristics of the geometric nonlinear sources, the sag effect of cables was

accounted for by the Ernst’s equivalent elastic modulus concept, the structural geometric change due to large displacement was included in updating each set of node

coordinates, whereas the axial force-bending moment interaction effect was considered

by the stability beam functions. Actually, by using the Ernst’s equivalent elastic modulus concept, the secant stiffness matrix of an inclined cable is simply equal to the

stiffness matrix of a truss element with length L and cross-sectional area A. Truss

elements can therefore be sufficiently used to model the inclined cables. Both bending

and axial forcing members such as the girder and towers are suitably modeled by

general beam elements. It can be assumed that the geometrical deformations of

structural members in a long-span cable-stayed bridge are characterized by large

displacement and small strains.

The material nonlinear analysis of a long-span cable-stayed bridge depends on the

nonlinear stress-strain behavior of individual materials for individual structural

elements. When some points (integration points) of an element exceed the yielding

limit of individual materials, the stiffness matrix of the element should be revised to

form the elastic plastic stiffness matrix.

For a long-span cable-stayed bridge, the dead loads always contribute the most to

total bridge loads. After the bridge is completed and before the live loads is applied,

the bridge has sustained large dead loads so that the large deformations and initial

stresses that already exist in each member should be considered. Therefore, the

geometrical nonlinear analysis of a long-span cable-stayed bridge under live loads

should start from the nonlinear equilibrium configuration after dead loads are applied.

A simplified 2D numerical model of the bridge is used for obtaining their nonlinear

responses and self-weight and live load distributions are considered. The bridges

studied are defined by combining characteristics such as the layout of the stays, fan or

harp pattern and the stiffness of the deck.

3. MATHEMATICAL MODEL OF THE STUDIED BRIDGES

3.1 Description of Cable-Stayed Bridge The example bridge studied here is Nile River long span cable-stayed bridge with

350 m central span length, 150 m left/right side spans. The elevation view of the bridge

is shown in Figure 1. The deck cross section is an aerodynamically shaped closed box

steel girder 18 m wide and 1.2 m high. The bridge towers are H-shaped hollow

reinforced concrete towers 105 m high. The four groups of cables are composed of

high strength steel wires 7 mm in diameter with from 294 to 1175 wires per cable. The

stay cables are single arrangements. The three main bridge elements of the example

bridge, namely, steel girder, cables, and reinforced concrete towers are composed of

two different materials. The weight per unit volume of each cable depends on the

number of wires in individual cables, the properties of each structure elements are

shown in Tables 1 and 2.


3.2 Reference Design A general assumption in cable supported bridge design is to have a reduced global

bending moment to about zero under self-weight loading. This means that one wants to

achieve that the self-weight load is completely supported by the cables. This can be

approximately achieved by manipulating the initial tensile force in the stayed cables

with minimum deck deflection and minimum bending stresses caused by the global

bending moment in the stiffening girder. In the FEM program this initial tensile force

on the main cable is done by applying a temperature load that causes the cables to

become shorter which is just a modeling tool to apply a pretension on a structural

member. The ideal state of a cable-stayed bridge can be defined as the minimized total

bending energy accumulated along the girder. The dominant issue of the design and

build of a cable-stayed bridge is to compute and achieve the ideal state. The dead and

traffic loads including impact on the girders are transmitted to the pylons by inclined

cable stays.

Harp System (a)



Figure 1 Layout of the cable stayed bridges

Table 1 The material properties for cable stayed bridge

Material Type Unit weight; KN/m3

Modulus of Elasticity; KN/m 2

1 Box steel Girder 76.973 199947978.8

2 Pylon 23.563 33500000.0

3 Cable 76.973 199947978.8

Table 2 The cables properties used to study the behavior of

Cable stayed bridge

Diameter, m Area of cross section; cm2 Weight; KN/m

0.12 113.04 0.8817

0.16 200.96 1.5675

0.20 314.00 2.4492

0.24 452.16 3.5269

The total width of the Nile River and river banks is succeeding 1000 m and the

average width of the river is 500 m. Horizontal navigation clearance of 350 m and

Vertical of 9.10 m are requested to be considered in design of the cable-stayed bridge.

The bridge has overall width of 18.0 m. A main span length of 350 m and two 150 m

side spans length with several approach spans are chosen as a starting point for the so-

called reference model.

To get the best configurations of cables and the optimum pylon height, many

parametric studied are taken into considerations. The major of these parameters are the

arrangements of cables, height of pylon to span ratios (H/L), the inertia of deck and

pylon. The design of a pier-to-deck connection should consider the control of both

longitudinal and transverse response. The critical response direction will depend on the

relative magnitude of seismic and eccentric gravity loads, as well as displacement

limits for the different response directions. For standard cable-stayed bridge

configurations, the connection in the longitudinal direction will need good stiffness to

limit deck displacements due to eccentric gravity loads.

Fan System (b)


The properties of the reference design of cable stayed bridge are that: the pylons are

composed of reinforced concrete with hollow rectangular uniform section 18m x 2m

giving inertia of Ix = 562.583 m4, Iy = 10.583 m

4 and Area A = 19 m

2. The deck is box

girder with an equivalent thickness of 0.2 m, with a width of 18 m and Iy = 37.903 m4,

Ix = 0.3127 m4 and Area A = 1.3403 m

2. The height of tower = 70 m and height of

pylon above the deck equal to 60 m. The connection between the pylon and deck is

rigid, while the pylon base is fixed and other two supports are rollers. The cables used

to study the behavior of cable stayed bridge have the properties shown as Table 1 and

with modulus of elasticity E=1.999 x 108 KN/m

2.

3.3 Finite Element Modeling This section introduces the finite element three-dimensional model of the studied

bridges. Figure 1 shows the configuration for the studied bridges and the number of

nodes and of cable elements in the global coordinate system. The girders, which have

the central span length of 350 m, are supported by a series of cables aligned in a fan-

type, and a harp-type structure respectively. The girders, the cross beams and the

pylons are modeled using a number of beam–column elements. Each cable stay

consists of an equivalent straight truss element, and is connected with pins at the

girders and the pylons. The behavior of cable-stayed bridge depends highly on the

manner in which the girders are connected to the pylons. In finite element analysis, the

girder and pylon are simulated by beam elements, while the cables by tension-only

truss elements. Nonlinear factors, large deformation, initial internal force and sag of

the cable are taken into account.

A three-dimensional nonlinear finite element model is developed for cable-stayed

bridges under static loadings based on the total Lagrangian formulation. The model can

account for the large displacements that are usually associated with extended in plane

contemporary cable-supported structures. At each step the equilibrium of the structure

is established taking into account displacements (geometric non-linearity) and laws of

behavior of the materials. In this paper; SAP2000 Ver. 14 program [11] is used for

nonlinear static analysis of the behavior of cable stayed bridge. This program enables

the designer to model a structure and to apply certain loads and loading combinations

from which the effects like member forces and deflections can be calculated. With

beam elements, the 3D model is built up with one dimensional line elements. This

enables to model the total bridge structure and calculate member forces due to certain

load cases and combinations. The scope of the model is to be able to analyze the model

statically in a three dimensional way. Also an assessment will be made with respect to

the geometric nonlinear effects of a cable supported bridge, the so called second order

effects.

For FEM modeling, the girder is divided into 130 beams-column elements and each

tower are divided into 13 beam-column elements. Each cable is treated as a plane truss

element. Because of the complex cross-section shape of the bridge, for simplicity, the

equivalent thin-walled box section of the girder and towers are used [12, 13]. The

equivalent sections are obtained by equalizing the cross-section areas and section

inertia moments of the girder and towers. The connection between the pylon and deck

is rigid, while the pylon base is fixed and other two supports are rollers as shown in

Figure 1. Under normal design loads, the material in a cable-stayed bridge is



considered to remain elastic; however, the overall load deformation relation can still

nonlinear. The analysis is carried out on an elastic model frame, making it possible to

take into account the influence of geometric non-linearity in members in compression

(second order effect) which are produced by finite deformations coupled with changes

in the stiffness of a structure under applied loadings.

3.4 Loading for Analysis The static analysis for all considered cases is carried out with uniformly distributed

dead; DL (self weight + super imposed load) and super imposed load; LL loads along

all spans lengths with intensity of 20 and 100 KN /m, respectively. The ultimate load

combination that is taken into consideration is Wu = 1.4 DL + 1.6 LL.

4. NUMERICAL ANALYSIS AND DISCUSSIONS

Parametric studies on cable-stayed bridges are performed for investigating the

individual influence of different key design parameters in such bridges. The geometry

of the bridges is defined on the basis of the parametric study; each one of the bridges

has a total length of 650 m spaced out in one principal central span of 350 m and two

side spans of 150m. The length of the spans, the layout of stays and the distance

between stays are detailed for both fan and harp patterns in Figure 1. The main

mechanical properties are detailed in Tables 1 and 2. Numerical models of the bridges

are developed and studied by means of linear static analysis using the finite element

code SAP2000 [11]. The deck and pylon of the bridge is modeled using beam-column

elements while truss elements are considered for the stays. The nonlinear behavior of

the stays is taken into account with Ernst's modulus of elasticity, the influence of

geometric non-linearity in members in compression (second order effect) which are

produced by finite deformations coupled with changes in stiffness of a structure under

applied loadings.

In this paper, different key design parameters of cable stayed bridge are studied to

get their influence on the principal characteristics of the target bridge, which are: the

mechanical properties and layout system of stayed cables; the mechanical properties

and height to span ration of the pylon; the mechanical properties of the deck. Layout

system of cables either fan or harp patterns and a wide range of cables’ cross section diameter from 012 m to 0.24 m are investigated. Moreover; the effect of moment of

inertia of deck variation as a ratio from 114.07% to 131.66% to that of reference

design, the moment of inertia variation of pylon from 113.61% to 127.32% to that of

reference design with different values of pylon height to span ratio H/L from 0.17 to

0.5 are studied. The static analysis is carried out taking into account the symmetrical

gravity load only and the effect of cables’ cross section diameter is studied for H/L

equal to 0.3 (H = 105 m, and L main span = 350 m, L left = L right = 150 m).

4.1 Effects of Mechanical Properties and Layout System of Stayed cables It can be seen that there is slight difference in the maximum deflection at

mid span of the deck between harp and fan cable layout system, moreover, it

is obviously that with cable diameter increases, the deflection for harp and fan

layout system decreases for stiff cable, and the difference in deflection

between harp and fan is reduced as shown in Figure 2. The variation of


diameter of cables from 0.12 m to 0.24 m could result in the bridge deck’s deformation response decrease in the harp system by percentage from 26.3%

to 51.4%, and in the fan system from 25.6% to 50.3%.

The bending moment peak response at the deck’s mid span for fan layout system displays bigger values than that for harp layout, this effect decreases

with the stiff cable system, could reach 7.3% for low stiffness of cable system.

The bending moment peak response decreases 20% with increasing the

diameter of cables from 0.12 m to 0.24 m as shown in Figure 3.

Figure 2 Maximum deflection variation with cables' diameter

at the deck’s mid span

Figure 3 Maximum bending moment variation with cables' diameter

at deck’s mid span



Figure 4 Normal force variation with cables' diameter at 150 m

distance on the deck

The normal force near the pylon (150 m distance on the deck) gets higher values

with stiff cable system, this effect reach around 19% and 28% for fan and harp systems,

respectively, as shown Figure 4. The influence of the stays layout is analyzed with the

same properties in the previous except of the height H above the deck is equal to 105 m

and diameter of stay equal to 0.2 m. while the cable layout system has no significant

effect on the deflection peak response at mid span of the deck, the deflection response

is significantly depends on the cable layout system in the side/middle span around

pylon, it can be seen that there is very little difference in the level of deformation at the

mid span but there are a gap between the response of the deflection between the fan

and harp system in the side span as shown in Figure 5. The bending moment response

of the deck is affected by the cable layout system in the region around the pylon deck

connection only as shown in Figure 6.


Figure 5 Deflection response of the deck

Figure 6 Bending moment response distribution over the deck



Figure 7 Normal force response distribution over the deck

It should be said that the deck is appreciably more heavily stresses in a bridge of

harp pattern than in fan pattern. In the particular case of this study, the difference is

about 50% under combination load. As a result, the harp layout appears to be less

suitable for large-span bridges, since the value of the normal force calls for

considerable strengthening of the cross-section, to provide both strength and stability in

addition, the total quantity of cables required is higher as shown in Figure 7.

The variation of bending moment response along the pylon height with

cable cross section's diameter from 0.12 m to 0.2 m in the harp and fan system

is shown in Figure 8, as the cable system gets stiffer as cable's diameter

increases, the bending moment response decreases around 17% and 23.5% at

the pylon base for harp and fan layout system, respectively. The harp layout

system has higher bending moment demands at pylon base compared to that

of fan layout system irrespective of stiffness level of cable system.


Figure 8 Bending moment response along the pylon height

Figure 9 Normal force response along the pylon height

The variation of normal force response along the pylon height with cable

cross section's diameter from 0.12 m to 0.2 m in the harp and fan system is

shown in Figure 9, as the cable system gets stiffer as diameter's increase, the

axial force response decreases around 12.5% and 10.5% at the pylon base for

harp and fan layout system, respectively. The fan layout system has higher

axial force demands along the pylon height compared to that of harp layout

system irrespective of stiffness level of the cable system. The normal force

distribution along the pylon height for fan layout system has almost constant



uniform distribution with values approach the axial force demands at pylon in

deck level that could affect the pylon stability due to buckling.

The cables of the single connection point solution are all relatively long in

comparison to the multiple connection point solution, in which the cables

close to the base of the piers are short. As such, a uniform longitudinal

displacement of the deck imposes much larger strains on the short cables,

which implies that the longitudinal displacement capacity of the bridge is

much lower than an identical bridge with a fan-type cable arrangement. While

the increased displacement capacity offered by the fan-arrangement may be

attractive, there are also reasons for which the distributed cable arrangement

of may be preferred. For example, the shorter cables will provide a stiffer

solution, which may be particularly useful in limiting the lateral displacement

of the deck, thereby limiting demands on dampers and expansion joints.

Kawashima et al. [14] also pointed out that greater damping could be expected

in the longitudinal mode when a distributed cable arrangement is adopted. As

such, in deciding on a cable arrangement, the designer should weigh the

benefits of reducing deformation demands on dampers and joints against the

increased cable diameters that are likely to be required to sustain the larger

design forces associated with the stiffer system.

4.2 Effects of Mechanical Properties and Height to span ratio of Pylon The effects of pylon height to span ratio; H/L range from 0.17 to 0.50 on

bridge response are studied, it is obviously that the increasing of H/L leads to

significant decrease of deflection response of the deck. The deflection

response decreases as H/L increases, these reductions in case of harp system

reach 25.5 % and 32.2 % for H/L equal to 0.30 and 0.50 compared to the

response for H/L= 0.17; respectively, while these reductions in case of fan

system reach 24.5% and 30.4% as shown in Figure 10.

Figure 11 shows that the bending moment response demands display slight decrease

as pylon height to span ratio; H/L increases. The bending moment response of the deck

at mid span decreases as H/L increases, these reductions in case of harp system reach

3.5 % and 2.3 % for H/L equal to 0.30 and 0.50 compared to the response for H/L=

0.17; respectively, while these reductions in case of fan system reach 5.8% . In brief,

the increase of the height of tower is beneficial to decrease the response, but larger

cross-section of tower and higher cost are needed.


Figure 10 Deflection response of the deck at mid span for

different H/L values

Figure 11 Bending moment response of the deck at mid span for

different H/L values

Figure 12 shows that the normal force response demands display

significant decrease as pylon height to span ratio; H/L increases. The normal

force response of the deck at pylon position (deck distance 150 m) decreases

as H/L increases, these reductions in case of harp system reach 39.4 % and

61.7 % for H/L equal to 0.30 and 0.50 compared to the response for H/L=

0.17; respectively, while this reductions in case of fan system reach 36.6 %

and 59.2% . The deck inertia of 37.9 m4 is used as a reference value, while the



pylon inertia is varied from 562.58 m4 to 715.75 m

4. The pylon rigidity has

slight effects on the deformation demands of the deck, the reduction

percentage is not more that 3.5 % either of harp or fan layout system for an

increase of 1.5 times of the pylon reference inertia as shown in Fig .13.

Figure 12 Normal force response of the deck at pylon (150 m deck distance)

for different H/L values

Figure 13 Deflection response of the deck at mid span for different pylon inertia



different pylon inertia

Figure 15 Normal force response of the deck at pylon for different pylon inertia

Figs. 14 and 15 show that the pylon rigidity has slight effects on the

bending moment/axial force demands of the deck, the reduction percentage is

not more than 1.6 % either of harp or fan layout system for an increase of 1.5

times of the pylon reference inertia. Under the dead load, with the increase of

the pylon height, the horizontal component of cable force decreases gradually.

In the meantime, the axial force and stress in girder and the anchorage force



also reduce gradually, with the increase of the tower height, the axial force in

girder, the stresses in cable and tower decrease. At the same time, the

horizontal component of anchorage force increases, while the vertical

component increases. In the case of deflection response, the value of deck

decreases, while that the value of the pylon increases a little. These results

indicate that the vertical rigidity increases while the longitudinal rigidity

decreases a little.

4.3 Effects of Mechanical Properties Bridge Deck The pylon inertia of 562.58 m

4; the height of pylon above the deck remains of 105

m (H/L = 0.3); cable stay diameter of 0.2 m and number of cables of 28 x 4 are used as

a reference values, while the deck inertia is varied from 37.9 m4 to 49.9 m

4. The deck

rigidity has significant effects on the deformation demands of the deck, Figure 16

shows this tendency of a decreasing deflection of the deck, the reduction percentage

reaches more that 47% and 50% for harp and fan layout system; respectively. So it can

concluded that the increasing of deck inertia plays a big role in decreasing deflection of

the bridge this underlines the importance of the role played by the stays.

Figure 16 Deflection response of the deck at mid span for different deck inertia

Figure 17 shows the bending moment response with the variation of deck rigidity,

where, the peak response of the bending moment demands at mid span of deck get

higher values as deck rigidity increases, and reach more than 85% and 89% for harp

and fan cable system, respectively. The bending response is accompanied by an

extension of the highly stressed zone of the deck. Figure 17 clearly shows that with an

increasing deck inertia/stiffness, the deck tends to carry a larger part of total bending

moment and smaller participating by the cable system and the deflection reduces

significantly. Increasing the bending stiffness has a significant effect on the moment

distribution. A larger stiffness of the deck means the bending moments increase


significantly. These observations lead to the conclusion that a deck with a high inertia

in the longitudinal direction is not basically favorable. It attracts considerable bending

moments, without appreciably reducing the forces in the pylons and the cables, and it

must be dimensioned in an appropriate manner.


different deck inertia

Figure 18 Normal force response of the deck at pylon for different deck inertia



The development of normal force peak response in the deck near the pylon (deck

distance 150 m and 500 m) as a function of the deck rigidity is shown in Figure 18. It

can be noticed that the significant increase of bending moment response is

accompanied by slight increase in the normal force response in the deck. The fan

layout cable system display higher normal force response compared to that of harp

system, the normal force response increases for deck inertia of 49.9 m4 by more than

7% and 14% of that of reference case, respectively.

5. CONCLUSIONS

Cable-stayed bridges have several advantages such as a wide range of span lengths

and arrangements, the possibility of limiting and eliminating piers, slender decks,

flexibility in the construction schedule, reduction of environmental impact (especially

for river crossings), increased traffic safety, and enhanced appearance. One of the main

advantages of these bridges is that they can be progressively erected on self-anchored

cantilevers without interfering with roads or rivers underneath the structure

This paper discussed some of the important key design considerations for cable-stayed

bridges under gravity loads. The mathematical model of a cable-stayed bridge is

formulated for single plane of cables with a global system of coordinates for bridges

having harp and fan shapes. All bridges have three spans of 150, 350, 150 m, each side

have twenty and eight cables in each side of the pylon. Different parameters studied to

get the influence of the principal characteristics, which are the layout of the stays, the

inertia of the deck and the pylons, and the change of diameter of the cross section of

cables. The recommendations of different cable-stayed bridge solutions have been

highlighted, with review of deck sections, pylon sections and height to span ratio and

cable arrangements and stiffness.

The investigations built on the factors that affect the behavior of the three span

cable stayed bridge in this research have led to the following conclusions: The

deflection of the deck is slight depends on the layout of cable system either harp or fan

system. The bending moment demands in the deck using the fan cable system are

higher than that in harp system. As the cable system gets stiffer, the normal force

demand in the deck gets higher, which could lead to instability due to buckling. The

deflection demands of the deck significantly decrease as the pylon height to span ratio;

H/L increases, while the normal force demands decreases. A deck with a high inertia in

the longitudinal direction is not basically favorable. It attracts considerable bending

moments, without appreciably reducing the forces in the pylons and the cables, and it

must be dimensioned in an appropriate manner. The pylon is appreciably more heavily

stressed in a bridge of the fan pattern than in one of harp pattern. In the particular case

of this study, the difference between normal in fan and harp is increasing when near to

the top of pylon 85.75% from the value of fan system at height 55 m from the deck and

reduce when near to the deck and there are a reversely effect on the change of stays on

the deck. As a result, the fan layout appears to be less suitable for large-span bridges.

For three span bridges the optimum side span length is 0.40 - 0.45 of the length of the

main span. The height of the tower above the deck is usually 0.2 times the length of the

main span. Close spacing of cables can reduce longitudinal bending moments in the

deck and the deck thickness. Regarding the force distribution and deflection, it is


favorable to consider a stiff cable system, to increase the global stiffness of the bridge

and to reduce the maximum bending moment in the girder. It is more profitable to

increase cable system axial stiffness in order to increase the stiffness of the bridge and

to reduce the bending moments in the girder. Designing and dimensioning structural

bridge members under bending is always less effective and more material consuming

than that of members under tensile loading, such as the cable system.

6. REFERENCES

[1] Troitsky M.S. (1988): Cable-Stayed Bridges: An Approach to Modern Bridge

Design, Nostrand Reinhold Co., N.Y.

[2] Gimsing N.J. (1997): Cable Supported Bridges Concept and Design; 2nd

ed, John

Wiley and Sons Inc, New York.

[3] Walther R., Houriet B., Isler W., Moia P. and Klein J. F. (1999): Cable Stayed

Bridges, 2nd

ed, Thomas Telford, London, U.K.

[4] Calvi G.M., Sullivan T.J. and Villani A. (2010): Conceptual Seismic Design of

Cable-Stayed Bridges, Journal of Earthquake Engineering; 14(8): 1139-1171.

[5] Chang F.K., Cohen E. (1981): Long-span bridges: state-of-the-art. Journal of

Structural Division, ASCE; 107(ST7): 1145–1160.

[6] Fleming J. F. (1979): Nonlinear static analysis of cable-stayed bridge structures,

Computers and Structures; 10(4): 621–635.

[7] Virlogeux M. (2001): Bridges with multiple cable-stayed spans, Structural

Engineering International; 11(1): 61–82.

[8] Nazmy A.S. and Abdel-Ghaffar A.M. (1990): Three-dimensional nonlinear static

analysis of cable-stayed bridges, Computers & Structures; 34(2): 257–271.

[9] Abdel Raheem S.E. and Hayashikawa T. (2003): Parametric study on steel tower

seismic response of cable-stayed bridges under great earthquake ground motion,

Structural Engineering and Earthquake Engineering, JSCE; 20(1): 25-41.

[10] Ernst J.H. (1965): Der E-Modul von Seilen unter berucksichtigung

desDurchhanges, Der Bauingenieur; 40(2): 52–55 (in German).

[11] Computer and Structures (CSI), Inc. (2011): SAP2000 Advanced 14.0.0 Software.

Structural Analysis Program, Berkeley, California.

[12] Ren W.X. (1997): Nonlinear static and seismic behavior of long span cable-stayed

bridges, Research Report No. 2, Department of Civil Engineering, Nagoya

Institute of Technology, Nagoya, Japan.

[13] Ren W.X. (1999): Ultimate behavior of long-span cable-stayed bridge, Journal of

Bridge Engineering, ASCE; 4(1): 30-37.

[14] Kawashima K., Unjoh S. and Tunomoto M. (1993): Estimation of damping ratio

of cable-stayed bridges for seismic design, Journal of Structural Engineering,

ASCE; 119(4): 1015–1031.

http://www.sciencedirect.com/science/journal/00457949

http://www.sciencedirect.com/science/journal/00457949/34/2



باري ل اخطي ي ا لسلوك ااستاتي مشدودةدراسة بارمترية ابات ا ب

باري ل اخطي ي ا سلوك ااستاتي دراسة ا بحث مشدودةيهدف هذا ا ملجمة( ا ابات )ا ا ب تحت تأثير حية ميتة واأحمال ا ي تطوير و من اأحمال ا تا اسق ودراسة جدوى وبا وبري تقديم تصميم مت هر موذج فوق

هدف، يل. من أجل تحقيق هذا ا معامات ا رئيسية تم عمل دراسة شاملة تصميم ا تي تلعب دورا مهما في ا واوبري تسلوك ا ك تصميم قديم وصياغة مجموعةوذ توصيات فيذ من ا ن ت عمل تصميم تم حيث . موذج يم

وب ابات ري مشدودمرجعي ي اخطي تحليل إجراءثم عمل تمثيل رياضي ب اصر باستخداماستاتي ع ظرية امحددة. تم دراسة لوص تأثيرا وبري يل و تغيير قيم معامات تصميم رئيسية علي سلوك ا ومن تصميم. أفضل إ

تي تم دراستها: معامات ا ابات، و ا ل/ترتيب اقطاع ا قيثارة، ش مط ا برج رتفاعسبة اابات إما مروحة أو اوبري برج، وااتصاات بين بحر/اتساع ا وبري ا ية ظهر ا ي ا مي خواص ا برج وا وبري وا قطاع ظهر ا

حامل. اقشة تم عرض ا سابقة علي على تأثيرتائج وم عوامل ا شد قيم ا هائي ا ابات ا ك في ا ذ قوى و ا

محو عزوم وقوى ريةا مختلفة ا ترخيم وقيم ا مختلفة واإزاحات ا اصر في ا ع ة ا و م شائية ا باري هذ اإ وأخيرا، امتعلقة تم صياغة تاجات ا ابات.بعض ااست مشدودة ب باري ا بتحليل وتصميم ا

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